I want to move my mother to another nursing home. Is there a charity that can help with this?

None that I know of, though there may be something locally. One thing ombudsmen do well is transitions. Try going to www.ltcombudsman.org and put in the Zip code of where she is now. Check with that person. You may want to check using the Zip code of the destination, as well.
Good luck,
Carol

PP:

Asking administration staff at the current NH might help too. I'm sure they have experience with this and could refer you to someone affordable. Wish you the best.

-- ED